Methods and apparatus to improve reach calculation efficiency

ABSTRACT

Methods, apparatus, systems and articles of manufacture are disclosed to improve reach calculation efficiency. An example method includes estimating, with a processor, a sample distribution of marketing data to generate a maximum entropy distribution, generating, with the processor, a geometric distribution based on estimating a minimum cross entropy of (a) the maximum entropy distribution and (b) the sample distribution of marketing data, and improving calculation efficiency of the public reach of the sample distribution of marketing data by generating, with the processor, conserved quantity expressions of the geometric distribution.

RELATED APPLICATION

This patent claims the benefit of, and priority to U.S. ProvisionalApplication Ser. No. 62/212,097, entitled “METHODS AND APPARATUS TOCALCULATE GROSS RATINGS POINTS AND REACH WITH ENTROPY,” which was filedon Aug. 31, 2015, and is hereby incorporated herein by reference in itsentirety.

FIELD OF THE DISCLOSURE

This disclosure relates generally to market data analysis, and, moreparticularly, to methods and apparatus to improve reach calculationefficiency.

BACKGROUND

In recent years, market analysts have measured observation impressionsassociated with media, in which impressions may include sales of anadvertised product, observations of an advertisement, observations of aparticular broadcast event, etc. To gain an understanding of aneffectiveness of a particular media distribution technique (e.g.,advertisements via television, advertisements via Internet media, etc.),the analysts typically calculate a gross rating point (GRP). The GRP iscalculated as a ratio of a number of observed impressions and a definedpopulation and, to account for unique impressions, the analysts may alsocalculate a corresponding reach metric, which is a ratio of a uniquenumber of exposures and the defined population.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic illustration of a negative binomial distributionmodel traditionally applied to published reach calculations.

FIG. 2 is an example feasibility region plot that identifies feasibilityregions for traditionally applied methods to calculate published reachvalues.

FIG. 3 is a schematic illustration of an example market data evaluatorsystem to improve reach calculation efficiency.

FIGS. 4-6 are flowcharts representative of example machine readableinstructions that may be executed to implement the example market dataevaluator system of FIG. 3 to improve reach calculation efficiency.

FIG. 7 is a chart of example solutions to determine a published reach asa function of raw reach.

FIG. 8 is a block diagram of an example processor platform structured toexecute the example machine readable instructions of FIGS. 4-6 toimplement the example market data evaluator system of FIG. 3.

DETAILED DESCRIPTION

In recent years, a gross rating point (GRP) metric, which is sometimesreferred to as an average or a mean, was calculated as a ratio of anumber of observed impressions and a defined population, but the GRPdoes not account for whether such impressions were unique to just oneindividual from the defined population. As such, a reach metric istypically used together with the GRP calculation to consider uniqueimpressions. The reach metric is calculated as a ratio of a uniquenumber of exposures and the defined population.

A market analyst typically measures both the GRP and the reachempirically for a particular event (e.g., an advertising campaign), butmay wish to understand a corresponding reach value in the event a newlyproposed (e.g., published) GRP value occurs. For example, a raw (e.g.,initial) GRP and raw reach value measured from panelist observations mayreveal values of 100 and 30, respectively. However, this example GRPvalue is associated with a particular event (e.g., advertising campaign)having a particular amount of advertising revenue applied thereto. Themarket analyst may know that one or more changes to the marketingcampaign can boost the GRP value from 100 to 200 (e.g., spending moreadvertising dollars to utilize a greater number of media outlets, suchas purchasing more commercial air-time, and/or purchasing commercialair-time during prime viewership times of day). Using the originallyprovided market data, the analyst applies a model to scale the GRP upfrom 100 to a value of 200 to determine a corresponding reach value atthe newly published GRP value.

Traditionally, the market analyst determines a corresponding publishedreach value from the proposed published GRP value by applying a negativebinomial distribution (NBD) model. The NBD models a frequencydistribution to reflect a percentage of the population that participatedin the impression(s), and assumes an infinite domain. When the NBD modelis applied to circumstances having a particular number of samples,accurate results may occur, however the NBD model has erroneous resultsfor certain feasibility regions, particularly when the number of samplesis relatively low or below a threshold value. In the event suchproblematic feasibility regions occur in view of the provided marketdata (frequency distribution), traditional techniques applied byanalysts include application of a Poisson distribution, which has becomean industry standard. Unfortunately, application of the Poissondistribution produces results that are independent of the published GRPand/or the original market data (frequency distribution). Stateddifferently, application of the NBD model presupposes a particulardistribution in an attempt to fit the data to that distribution, andpredictions are derived therefrom.

Examples disclosed herein prevent any assumption and/or presuppositionof which distribution to use and, instead, preserve the integrity of theknown marketing data to derive a distribution that best fits that data.Further, examples disclosed herein apply a principle of maximum entropyand minimum cross entropy to solve for the proper distribution that bestfits the market data. Entropy reflects disorder in a distribution, andexamples disclosed herein apply the principle of maximum entropy toderive the most accurate distribution in an effort to reduceuncertainty. Because a maximum entropy exposes what is maximallyuncertain about what is unknown, any other distribution, such as anassumed NBD distribution, means that more information is being forcedinto that distribution/model that does not reflect the empirical data(e.g., the provided market data). Examples disclosed herein determine amaximum entropy of which observations are at zero (e.g., zeroimpressions), which is a portion of the NBD model that produces the mosterror. Additionally, examples disclosed herein apply the minimum crossentropy to modify the maximum entropy distribution to create a newdistribution that still depends on values from the empiricalobservations, unlike the application of the Poisson distribution.

As described above, GRP is a metric to measure impressions in relationto the number of people in target data for an advertising campaign,which is calculated in a manner consistent with example Equation 1.

$\begin{matrix}{{GRP} = {100*{\frac{Impressions}{{Defined}\mspace{14mu}{Population}}.}}} & {{Equation}\mspace{14mu} 1}\end{matrix}$Additionally, as described above, reach is the total number of differentpeople in the defined population exposed, at least once, to the campaignduring a given period, which is calculated in a manner consistent withexample Equation 2.

$\begin{matrix}{{Reach} = {100*{\frac{{Unique}\mspace{14mu}{People}\mspace{14mu}{Exposed}}{{Defined}\mspace{14mu}{Population}}.}}} & {{Equation}\mspace{14mu} 2}\end{matrix}$A mathematical constraint between a GRP calculation and a correspondingreach calculation is that the GRP value must always be greater-than orequal to the reach value. For instance, if five (5) impressions aredetected and/or otherwise observed in a distribution having a populationof ten (10) people, then a corresponding GRP value is fifty (50).Because reach reflects an indication of how many unique exposuresoccurred, the number of unique exposures mathematically cannot exceedthe example five (5) impressions detected.

In the event the analyst can improve (e.g., increase) the GRP of theoriginal distribution by applying one or more marketing drivers(sometimes referred to as “scaling-up”), then the analyst would alsolike to justify whether such marketing drivers will be effective and/ora degree by which such marketing drivers will be effective bydetermining a corresponding reach value associated with the scaled-upGRP value. Traditional application of the NBD model have been used bymarket analysts under the qualified assumption that a scaled-upfrequency distribution follows the curve/distribution of that NBD model,as shown in FIG. 1.

In the illustrated example of FIG. 1, an NBD curve 100 includes anx-axis of GRP values 102 and a y-axis of probability 104. As describedabove, because the GRP reflects an average, the non-zero reachprobabilities 106 correspond to an example GRP of 1.33 (or 133 aftertraditional multiplication of the ratio of example Equation 1 by 100)108. The reach corresponding to the example GRP of FIG. 1 is 64 (or 64%)to reflect unique or different exposures of the audience. Because theNBD curve 100 includes all participants of a given defined population(e.g., the denominator of example Equations 1 and 2), a correspondingprobability at zero is known to be 36% (100−64). The NBD in the contextof GRP and reach is defined in a manner consistent with example Equation3.

$\begin{matrix}{{P(i)} = {\frac{\Gamma\left( {k + i} \right)}{{\Gamma(k)}{\Gamma\left( {i + 1} \right)}}*\left( \frac{1}{\left( {1 + a} \right)} \right)^{k}*{\left( \frac{a}{1 + a} \right)^{i}.}}} & {{Equation}\mspace{14mu} 3}\end{matrix}$In the illustrated example of Equation 3, P(i) reflects a proportion ofviewing an i^(th) spot/index (e.g., advertisement), Γ reflects a Gammafunction, and a and k are parameters of the model. As described above,application of the NBD model presupposes a particulardistribution/shape, to which provided observation data (e.g., marketdata) is forced. When the market analyst scales-up a given distributionhaving a new/published GRP value, the NBD is solved for a proportion ofzero (0) viewing spots (i.e., P(0)), in which the NBD equation ofEquation 3 simplifies to example Equation 4.

$\begin{matrix}{{P(0)} = {\left( \frac{1}{1 + a} \right)^{k} = {1 - {\frac{{Raw}\mspace{14mu}{Reach}}{100}.}}}} & {{Equation}\mspace{14mu} 4}\end{matrix}$Parameters a and k are isolated in a manner consistent with exampleEquation 5.

$\begin{matrix}{{{Expected}\mspace{14mu}{Value}} = {{a*k} = {\frac{{Raw}\mspace{14mu}{GRP}}{100}.}}} & {{Equation}\mspace{14mu} 5}\end{matrix}$Under the assumption that any scaled-up distribution will reasonablyconform to the NBD, the new/published/target GRP is scaled up togenerate a modified parameter a, which is derived in example Equation 6as scaled parameter A.

$\begin{matrix}{A = {a*{\frac{{Published}\mspace{14mu}{GRP}}{{Raw}\mspace{14mu}{GRP}}.}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$In connection with scaled parameter A, a new published reachcorresponding to the published GRP is calculated in a manner consistentwith example Equations 7 and 8.

$\begin{matrix}{{P(0)} = {\left( \frac{1}{1 + A} \right)^{k} = {1 - {\frac{{Published}\mspace{14mu}{Reach}}{100}.}}}} & {{Equation}\mspace{14mu} 7} \\{{P(i)} = {\frac{\Gamma\left( {k + i} \right)}{{\Gamma(k)}{\Gamma\left( {i + 1} \right)}}*\left( \frac{1}{1 + A} \right)^{k}*{\left( \frac{A}{1 + A} \right)^{i}.}}} & {{Equation}\mspace{14mu} 8}\end{matrix}$

To illustrate limitations of the traditionally applied NBD model, assumean observed distribution includes a GRP value of 100 with acorresponding reach value of 80. Applying example Equation 5 and solvingfor parameters a and k results in a value of −0.647 for parameter a, anda value of −1.546 for parameter k. For varying values of index (i),corresponding probability values result, as shown in Table 1 below.

TABLE 1 i P(i) 0 0.200 1 0.567 2 0.283 3 −0.079 4 0.052 5 −0.047As shown in the illustrated example of Table 1, application of the NBDmodel breaks down for the 3^(rd) and the 5^(th) index values, whichproduce erroneous negative values for probability. The qualifiedassumptions made by analysts using the NBD model described above includeat least two considerations, the first of which is that the applied raw(e.g., initial) GRP is greater than or equal to the reach. Thisconsideration is met by the example above, where GRP is 100 and thecorresponding reach is 80. However, the second consideration includes aninherent limitation of the NBD model for those combinations of GRP andreach that are not themselves mathematically impossible, but nonethelessimpossible when employed with the NBD model.

FIG. 2 illustrates a feasibility region plot 200 associated with the NBDmodel. In the illustrated example of FIG. 2, the feasibility region plot200 includes an x-axis to reflect reach values 202 and a y-axis toreflect GRP values 204. The example feasibility region plot 200 alsoincludes a mathematically impossible region 206, in which the GRP is notgreater than or equal to the reach. However, the example feasibilityregion plot 200 also includes an NBD-impossible region 208, which mayreflect valid values for GRP and reach that could be observed in marketdata, but cause the NBD model to fail, as shown by the results ofexample Table 1 above. Finally, the example feasibility region plot 200includes an NBD-feasible region 210 for which corresponding values ofGRP and reach will not cause the NBD model to fail.

When an analyst identifies circumstances where combinations of GRP andreach reside within the example NBD-impossible region 206, suchtraditional solutions apply a Poission distribution. In some examples,the Poission distribution is applied as a work-around to theNBD-impossible region 206 when the corresponding a parameter iscalculated at a value less than zero. The Poission distribution replacesthe NBD approach and is used with an alternate parameter lambda (λ) asshown in example Equation 9.

$\begin{matrix}{\lambda = {\frac{{Published}\mspace{14mu}{GRP}}{100}.}} & {{Equation}\mspace{14mu} 9}\end{matrix}$The published reach is then calculated in a manner consistent withexample Equation 10 for a proportion of viewing zero (0) spots(advertisements).

$\begin{matrix}{{{Proportion}\mspace{14mu}{Viewing}\mspace{14mu} 0\mspace{14mu}{Spots}} = {e^{- \lambda} = {1 - {\frac{{Published}\mspace{14mu}{Reach}}{100}.}}}} & {{Equation}\mspace{14mu} 10}\end{matrix}$To determine a corresponding frequency distribution for proportionsviewing any number of non-zero spots, example Equation 11 is applied.

$\begin{matrix}{{{Proportion}\mspace{14mu}{Viewing}\mspace{14mu} i\mspace{14mu}{Spots}} = {\frac{\lambda^{i}e^{- \lambda}}{i!}.}} & {{Equation}\mspace{14mu} 11}\end{matrix}$While application of the Poission distribution as shown above has beenadopted as an industry standard, worth noting is that this approach nolonger reflects any influence of the published GRP and completelyignores the original sample distribution.

Examples disclosed herein do not presuppose a distribution and attemptto fit the market data thereto, but rather generate one or morealternate distributions based on the available data. FIG. 3 illustratesan example market data evaluator (MDE) system 300 that includes an MDE302 communicatively connected to one or more campaign data sources 304via an example network 306. In the illustrated example of FIG. 3, theMDE 302 includes a marketing data interface 308, an example gross ratingpoint (GRP) engine 310, an example reach engine 312, an examplesimulation engine 314, an example conserved quantity engine 316, anexample maximum entropy (ME) engine 318, and an example minimum crossentropy (MCE) engine 320. In the illustrated example of FIG. 3, the MEengine 318 includes an example maximum entropy (ME) constraint manager322 and an example maximum entropy (ME) distribution evaluator 324. Inthe illustrated example of FIG. 3, the MCE engine 320 includes anexample MCE constraint manager 326 and an example published GRP manager328.

In operation, the example marketing data interface 308 retrieves adataset having an unknown distribution, such as market data associatedwith one or more of the campaign data sources 304. The dataset mayrepresent market behavior activity such as, but not limited to,impression count data associated with promotional activity in whichparticular participants of a population are exposed to advertisements.The example GRP engine 310 calculates a corresponding GRP valueassociated with the dataset in a manner consistent with example Equation1, and the example reach engine 312 calculates a corresponding reachvalue in a manner consistent with example Equation 2. In some examples,GRP and reach values that have been calculated from empirical datasetsare referred to as raw GRP values and raw reach values. On the otherhand, in the event an analyst wishes to use the empirical dataset toscale-up to a proposed/candidate published GRP value, which is not basedon empirically collected data, then traditional techniques employed theNBD model, as described above.

The example simulation engine 314 selects a candidate published GRPvalue of interest as a scaling-up value, and the maximum entropy engine318 estimates a distribution using the principal of maximum entropy togenerate a maximum entropy distribution. Generally speaking, entropyquantifies an uncertainty involved in predicting a value of a randomvariable. Entropy reflects an expected value (E) of information (X), asshown by example Equation 12.

$\begin{matrix}\begin{matrix}{{H(X)} = {E\left\lbrack {I(X)} \right\rbrack}} \\{= {\sum\limits_{i}\;{{P\left( x_{i} \right)}{I\left( x_{i} \right)}}}} \\{= {- {\sum\limits_{i}\;{{P\left( x_{i} \right)}\log_{b}{{P\left( x_{i} \right)}.}}}}}\end{matrix} & {{Equation}\mspace{14mu} 12}\end{matrix}$The principle of maximum entropy states that a probability distributionthat best represents the current state of knowledge is the one havingthe largest entropy. While an analyst goal may be to reduce uncertaintyin any prediction made, examples disclosed herein first establish afoundation of being maximally uncertain about what we do not know.Because entropy is maximum when all outcomes are equally likely, anyinstance that deviates from equally likely outcomes reduces the entropyby introducing information to a distribution.

The example ME constraint manager 322 establishes constraints for theprovided distribution to prevent computational waste, as shown byexample Equations 13 and 14.

$\begin{matrix}{{\sum\limits_{k = 0}^{\infty}\; q_{k}} = 1.} & {{Equation}\mspace{14mu} 13} \\{{\sum\limits_{k = 0}^{\infty}\;{kq}_{k}} = {u_{1}.}} & {{Equation}\mspace{14mu} 14}\end{matrix}$In the illustrated examples of Equations 13 and 14, three constraintsinclude (a) q₀ is a known constraint (also reflects a shorthand notationfor a probability at zero, which is 1-reach), (b) sums are expected toreach 100%, and μ₁ is an expected value mean, which is sometimesreferred to as a sample GRP or the empirically known GRP. Additionally,q_(k) reflects a reach value at a k^(th) frequency. At least oneadditional benefit of the example ME constraint manager 322 establishingthe constraints in a manner consistent with example Equations 13 and 14is that only positive values result for a probability set. This alsoensures that any impossibility region(s) will not affect predictionaccuracy.

The example ME distribution evaluator 324 applies the principle ofmaximum entropy to the distribution in a manner consistent with exampleEquation 15.

$\begin{matrix}{{{{maximize}\mspace{14mu} Q}->H} = {- {\sum\limits_{k = 0}^{\infty}\;{q_{k}{{\log\left( q_{k} \right)}.}}}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$In the illustrated example of Equation 15, Q reflects a reachdistribution with a maximized entropy (H). Example Equation 15 is solvedto generate a zero-modified geometric distribution using the knownvalues for the sample reach for a probability at zero (q₀) and thesample GRP (μ₁). In particular, the example ME constraint manager 322facilitates (e.g., based on the constraints of (a) μ₁, (b) q₀, (c) μ₂and (d) the sum equaling 100%) cross entropy calculations in a mannerconsistent with example Equations 16, 17 and 18, in which the crossentropy calculations consider a published GRP that may be achieved bythe market analyst.

$\begin{matrix}{{q_{k} = {{{Cr}^{k}\mspace{14mu}{where}\mspace{14mu} k} = 1}},2,\ldots\mspace{14mu},{\infty.}} & {{Equation}\mspace{14mu} 16} \\{C = {\frac{\left( {1 - q_{0}} \right)^{2}}{u_{1} + q_{0} - 1}.}} & {{Equation}\mspace{14mu} 17} \\{r = {\frac{u_{1} + q_{0} - 1}{u_{1}}.}} & {{Equation}\mspace{14mu} 18}\end{matrix}$The illustrated example of Equation 16 reflects the prior distributionand in example Equation 17, C reflects one of two constants tofacilitate cross entropy calculations, and in the illustrated example ofEquations 16, 17 and 18, r reflects the second of two constants tofacilitate cross entropy calculations of the geometric maximum entropydistribution.

The example MCE engine 320 modifies the maximum entropy distributiongenerated by the ME engine 318 with the principle of minimum crossentropy to create a new distribution that is constrained by thecandidate published GRP of interest. In particular, the example MCEconstraint manager 326 establishes constraints for applying the minimumcross entropy as shown by example Equations 19 and 20.

$\begin{matrix}{{\sum\limits_{k = 0}^{\infty}\; p_{k}} = 1.} & {{Equation}\mspace{14mu} 19} \\{{\sum\limits_{k = 0}^{\infty}\;{kp}_{k}} = {u_{2}.}} & {{Equation}\mspace{14mu} 20}\end{matrix}$Note that example Equations 19 and 20 are similar to example Equations13 and 14, but the former reflect constraints of reach values (p) atdifferent frequencies (k) and the published/target/candidate GRP (u₂).

The example MCE engine 320 applies a minimum cross entropy functionbased on (a) the maximum entropy distribution (Q) and (b) the unknowndistribution to be solved (P), as shown in example Equation 21.

$\begin{matrix}{{{{minimize}\mspace{14mu} P}->{D\left( {P\text{:}Q} \right)}} = {{p_{0}{\log\left( \frac{p_{0}}{q_{0}} \right)}} + {\sum\limits_{k = 1}^{\infty}\;{p_{k}{{\log\left( \frac{p_{k}}{{Cr}^{k}} \right)}.}}}}} & {{Equation}\mspace{14mu} 21}\end{matrix}$In the illustrated example of Equation 21, q₀ reflects the sample reachat a probability of zero, p₀ reflects the published reach at aprobability of zero, constraint values C and rare known from before, andp_(k) is the unknown reach values at the k^(th) frequency to solve for.In some examples, the minimum cross entropy is calculated as a measureof the difference between two probability distributions in a mannerconsistent with Kullback-Leibler (KL) divergence. The KL divergence is ameasure of information gained between distributions. The example MCEengine 320 integrates assistance values with zero-modified geometricconstraints in a manner consistent with example Equations 22 and 23.p ₀ =s ₀ q ₀   Equation 22.p _(k) =s ₀ s ₁ ^(k) Cr ^(k) where k=1,2, . . . ,∞   Equation 23.In the illustrated examples of Equations 22 and 23, s₀ and s₁ are solvedto satisfy the constraints, as described in further detail below, andexplicitly identified values of p₀ will guarantee a unique solution. Oneor more desired values for p_(k) can be solved by way of exampleEquations 16-18.

As described above, solving for s₁ to facilitate an integration of thepublished GRP constraint employs the example published GRP manager 328,which solves s₁ in a manner consistent with example Equation 24.

$\begin{matrix}{1 = {{\sum\limits_{k = 0}^{\infty}\; p_{k}} = {{p_{0} + {\sum\limits_{k = 1}^{\infty}\;{\frac{p_{0}}{q_{0}}s_{1}^{k}{Cr}^{k}}}} = {\left. {p_{0} - \frac{{Cp}_{0}{rs}_{1}}{q_{0}\left( {{rs}_{1} - 1} \right)}}\Rightarrow s_{1} \right. = {\frac{{p_{0}q_{0}} - q_{0}}{r\left( {{- {Cp}_{0}} + {p_{0}q_{0}} - q_{0}} \right)}.}}}}} & {{Equation}\mspace{14mu} 24}\end{matrix}$

With all of the constraints and assistance values solved (including s₁),the example published GRP manager 328 integrates the published GRPconstraints in a manner consistent with example Equation 25.

$\begin{matrix}{u_{2} = {{\sum\limits_{k = 0}^{\infty}\;{kp}_{k}} = {{0 + {\sum\limits_{k = 1}^{\infty}\;{k\frac{p_{0}}{q_{0}}s_{1}^{k}{Cr}^{k}}}} = {{\sum\limits_{k = 1}^{\infty}\;{k\frac{p_{0}}{q_{0}}\left( \frac{{p_{0}q_{0}} - q_{0}}{r\left( {{- {Cp}_{0}} + {p_{0}q_{0}} - q_{0}} \right)} \right)^{k}{Cr}^{k}}} = {- {\frac{\left( {q_{0} - 1} \right)\left( {{Cq}_{0} + {p_{0}\left( {- q_{0}} \right)} - p_{0}} \right)}{{Cq}_{0}}.}}}}}} & {{Equation}\mspace{14mu} 25}\end{matrix}$

The example conserved quantity engine 316 further simplifies to exampleEquations 26, 27 and 28 to solve for the published reach value toillustrate that quantities can be conserved for values of raw GRP, rawreach, published GRP and published reach.

$\begin{matrix}{u_{2} = {\frac{\left( {p_{0} - 1} \right)\left( {{p_{0}\left( {{u_{1}q_{0}} + q_{0} - 1} \right)} - {q_{0}\left( {u_{1} + q_{0} - 1} \right)}} \right)}{{p_{0}\left( {q_{0} - 1} \right)}^{2}}.}} & {{Equation}\mspace{14mu} 26} \\{\mspace{79mu}{\frac{\left( {u_{1} + q_{0} - 1} \right)q_{0}}{1 - q_{0}} = {\frac{\left( {u_{2} + p_{0} - 1} \right)p_{0}}{1 - p_{0}}.}}} & {{Equation}\mspace{14mu} 27} \\{\mspace{79mu}{\frac{\left( {\frac{u_{1}}{1 - q_{0}} - 1} \right)}{\left( \frac{1 - q_{0}}{q_{0}} \right)} = {\frac{\left( {\frac{u_{2}}{1 - p_{0}} - 1} \right)}{\frac{1 - p_{0}}{p_{0}}}.}}} & {{Equation}\mspace{14mu} 28}\end{matrix}$

From example Equation 28, because the form of the equation is similar onboth sides, and each individual side only contains variables of eitherthe prior and unknown, the example conserved quantity engine 316generates conserved quantity equivalent expressions. In particular, ureflects a mean (M), which is also referred to as a GRP, R reflects areach value, which can also be expressed as 1−q₀, and F reflects afrequency, which can also be expressed as a ratio of the mean and reach(e.g., u/1−q₀). An example equivalent expression for mean (M) and reach(R) generated by the example conserved quantity engine 316 is shown inexample Equation 29.

$\begin{matrix}{\frac{\left( {M - R} \right)\left( {1 - R} \right)}{R^{2}}.} & {{Equation}\mspace{14mu} 29}\end{matrix}$An example equivalent expression for mean (M) and frequency (F)generated by the example conserved quantity engine 316 is shown inexample Equation 30.

$\begin{matrix}{\frac{\left( {F - 1} \right)\left( {F - M} \right)}{M}.} & {{Equation}\mspace{14mu} 30}\end{matrix}$An example equivalent expression for reach (R) and frequency (F)generated by the example conserved quantity engine 316 is shown inexample Equation 31.

$\begin{matrix}{\frac{\left( {F - 1} \right)\left( {1 - R} \right)}{R}.} & {{Equation}\mspace{14mu} 31}\end{matrix}$

To illustrate a computational improvement of using the conservedquantities of derived example Equations 29-31 over the relatively morecomputationally burdensome NBD approach, in which a simultaneous systemof equations must be solved, assume an initial raw GRP value of 350 anda corresponding reach of 70. The raw GRP and reach values are derivedfrom empirically obtained market data. Also assume that a campaignmanager, market researcher, or other individual/entity determines that afuture campaign could be initiated to result in a published/new GRPvalue of 400, for which empirically obtained market data is notavailable. Assuming that the market researcher first employed thetraditional NBD approach, a computationally intensive system ofnonlinear equations would need to be solved in a manner consistent withexample Equations 4 and 5 described above. Parameters of the NBD modelresult in a=5.393285 and k=0.648955. Knowing the value for parameter k,scaled-up parameter A (or a₂) can be determined in view of the publishedGRP value to yield a value of 6.163. Finally, knowing A and k, thenew/published reach value can be solved in a manner consistent withexample Equation 7 to yield a new published reach of 72.1%.

However, taking the same initial raw GRP value of 350 and correspondinginitial raw reach value of 70, with a published GRP of 400 expected as aviable target for a future campaign, examples disclosed hereinfacilitate calculation via closed-loop conserved quantity expressionsthat yield substantially similar results. In particular, exampleEquation 29 is shown below using the GRP and reach values of interest,expressed as example Equation 32.

$\begin{matrix}{\frac{\left( {M_{1} - R_{1}} \right)\left( {1 - R_{1}} \right)}{R_{1}^{2}} = {\frac{\left( {M_{2} - R_{2}} \right)\left( {1 - R_{2}} \right)}{R_{2}^{2}}.}} & {{Equation}\mspace{14mu} 32}\end{matrix}$In the illustrated example of Equation 32, M₁ reflects the raw GRP of350, R₁ reflects the raw reach of 70, M₂ reflects the published GRP of400, and R₂ reflects the published reach, which is solved below andshown as example Equation 33.

$\begin{matrix}{\frac{\left( {3.50 - 0.70} \right)\left( {1 - 0.70} \right)}{(0.70)^{2}} = {\frac{\left( {4.00 - R_{2}} \right)\left( {1 - R_{2}} \right)}{R_{2}^{2}}.}} & {{Equation}\mspace{14mu} 33}\end{matrix}$Solving for R2 in example Equation 33 yields a published reach of 72.5%,which is substantially the same value as determined via the relativelymore computationally complex NBD approach. Additionally, by using theconserved quantity equivalent expression approach disclosed herein,problems associated with the NBD are avoided, such as effects of theNBD-impossibility regions, assumptions of a distribution shape being anappropriate fit for unknown data and/or resorting to a Poissiondistribution, which pays no regard to original distribution inputs.

Examples disclosed above consider an upper bound that is infinite, whileexamples disclosed below assume a maximum number of “spots” (e.g.,television spots, advertising spots, Internet placement spots, etc.)that can be used in a particular calculation. As such, any GRP value hasa maximum consistent with example Equation 34.Reach≤GRP≤N*Reach   Equation 34.In the illustrated example of Equation 34, N represents a finite numberof spots in which the left-hand side of Equation 34 must be true whilethe right-hand side of Equation 34 is assumed. For example, in the eventit is assumed a reach of 100%, then everybody is watching (e.g.,watching television), and the inequality reduces to example Equation 35.1≤GRP≤N   Equation 35.In the illustrated example of Equation 35, the left-hand side of theinequality states that viewers watch at least one spot, and theright-hand side of the inequality illustrates that every viewer iswatching every possible spot. Examples described below account for abounded GRP value by a numerical solution.

Similar to examples disclosed above, parameters from the NBD for the rawreach and GRP are calculated and scaled-up to that the expected value ofthe new NBD matches the published reach. However, for a finite number ofspots, the expected value of the first N spots may not equal thepublished reach value, thus the distribution is both truncated andmodified so that the domain is the first N spots and the expected valuewithin those N spots equals the published reach. To illustrate, aninterim GRP is defined in a manner consistent with example Equation 36.

$\begin{matrix}{{{Interim}\mspace{14mu}{GRP}} = {\sum\limits_{k = 1}^{N}\;{{{kf}(k)}.}}} & {{Equation}\mspace{14mu} 36}\end{matrix}$The interim GRP is employed to define a scale factor in a mannerconsistent with example Equation 37.

$\begin{matrix}{{Factor} = {\frac{{Published}\mspace{14mu}{GRP}}{{Interim}\mspace{14mu}{GRP}}.}} & {{Equation}\mspace{14mu} 37}\end{matrix}$For frequencies greater-than or equal to one (k≥1), each proportion ismultiplied by the factor in a manner consistent with example Equation38.g(k)=(Factor)f(k)   Equation 38.The zero frequency is then defined in a manner consistent with exampleEquation 39.

$\begin{matrix}{{g(0)} = {1 - {\sum\limits_{k = 1}^{N}\;{{g(k)}.}}}} & {{Equation}\mspace{14mu} 39}\end{matrix}$Accordingly, the published reach is represented in a manner consistentwith example Equation 40.Published Reach=1−g(0)   Equation 40.

The above examples take into account a finite value of N and results inthe expected value equal to the published GRP, but the raw and publishedvalues are treated as two different distributions. Improved approachesinclude assuming both distributions are truncated NBDs and fitting thedata to both. As described above in connection with example Equations13-15, an unknown distribution (X) is estimated using maximum entropygiven the constraint that E[X]=μ₁ and q₀=P(X=0). However, unlike exampleEquations 13-15 above, the domain is specified in a finite manner to be{0, 1, . . . , N}.

Additionally, finite domain examples include creating a new distributionwith minimum cross entropy in a manner similar to example Equations19-21, in which the domain is again specified in a finite manner to be{0, 1, . . . , N}. Numerical solution of the published reach (p0) may besolved in a manner consistent with example Table 2, in which the answerdepends only upon (a) a raw GRP value, (b) a raw reach value and (c) apublished GRP value. In particular, example Table 2 illustrates pseudocode that may be employed by the example MDE system 300. The examplepseudo code of Table 2 facilitates handling both distributions as afinite domain to find the best distribution that fits the known data.

TABLE 2 Raw_GRP=50; % (μ₁) Raw_Reach=30; % (q0) Published_GRP=200 % (μ₂)N=20; % Maximum number of “spots” % Construct constraints finite domainMaximum Entropy C=[ones(1,N+1)]; 0:N; 1 zeros (1, N)]; D=[1;(Raw_GRP/100); 1−(Raw_Reach/100)]; %Solve for prior distributionQ=MaxEnt(C,D); %New constraints C=[ones(1,N+1);  0:N]; D=[1,(Published_GRP/100)]; %Solve for prior distribution P=MaxEnt(C,D,Q);Published_Reach=100*(1−P(1));In the illustrated example of Table 2, variable C represents aconstraint matrix, not to be confused with variable C in exampleEquation 17, which represents a constant. Additionally, the pseudo code“P=MaxEnt(C, D, Q)” represents solving for the minimum cross entropysolution. As described in the illustrated example of FIG. 7, solutionsto the question of how a published reach depends on a function of rawreach is shown in view of an (a) NBD approach, (b) a maximum entropyapproach, (c) an NBD approach with finite correction (e.g., the industrystandard), and (d) a maximum entropy approach with a finite domain(e.g., via numerical analysis consistent with example pseudo code ofTable 2).

While an example manner of implementing the market data evaluator (MDE)of FIG. 3 is illustrated in FIG. 3, one or more of the elements,processes and/or devices illustrated in FIG. 3 may be combined, divided,re-arranged, omitted, eliminated and/or implemented in any other way.Further, the example marketing data interface 308, the example GRPengine 310, the example reach engine 312, the example simulation engine314, the example conserved quantity engine 316, the example maximumentropy engine 318, the example minimum cross entropy engine 320, theexample maximum entropy constraint manager 322, the example maximumentropy distribution evaluator 324, the example minimum cross entropyconstraint manager 326, the example published GRP manager 328 and/or,more generally, the example market data evaluator (MDE) 302 of FIG. 3may be implemented by hardware, software, firmware and/or anycombination of hardware, software and/or firmware. Thus, for example,any of the example marketing data interface 308, the example GRP engine310, the example reach engine 312, the example simulation engine 314,the example conserved quantity engine 316, the example maximum entropyengine 318, the example minimum cross entropy engine 320, the examplemaximum entropy constraint manager 322, the example maximum entropydistribution evaluator 324, the example minimum cross entropy constraintmanager 326, the example published GRP manager 328 and/or, moregenerally, the example market data evaluator (MDE) 302 of FIG. 3 couldbe implemented by one or more analog or digital circuit(s), logiccircuits, programmable processor(s), application specific integratedcircuit(s) (ASIC(s)), programmable logic device(s) (PLD(s)) and/or fieldprogrammable logic device(s) (FPLD(s)). When reading any of theapparatus or system claims of this patent to cover a purely softwareand/or firmware implementation, at least one of the example marketingdata interface 308, the example GRP engine 310, the example reach engine312, the example simulation engine 314, the example conserved quantityengine 316, the example maximum entropy engine 318, the example minimumcross entropy engine 320, the example maximum entropy constraint manager322, the example maximum entropy distribution evaluator 324, the exampleminimum cross entropy constraint manager 326, the example published GRPmanager 328 and/or, more generally, the example market data evaluator(MDE) 302 of FIG. 3 is/are hereby expressly defined to include atangible computer readable storage device or storage disk such as amemory, a digital versatile disk (DVD), a compact disk (CD), a Blu-raydisk, etc. storing the software and/or firmware. Further still, theexample MDE 302 of FIG. 3 may include one or more elements, processesand/or devices in addition to, or instead of, those illustrated in FIG.3, and/or may include more than one of any or all of the illustratedelements, processes and devices.

Flowcharts representative of example machine readable instructions forimplementing the MDE system 300 of FIG. 3 are shown in FIGS. 4-6. Inthese examples, the machine readable instructions comprise a program forexecution by a processor such as the processor 812 shown in the exampleprocessor platform 800 discussed below in connection with FIG. 8. Theprogram(s) may be embodied in software stored on a tangible computerreadable storage medium such as a CD-ROM, a floppy disk, a hard drive, adigital versatile disk (DVD), a Blu-ray disk, or a memory associatedwith the processor 812, but the entire program and/or parts thereofcould alternatively be executed by a device other than the processor 812and/or embodied in firmware or dedicated hardware. Further, although theexample program(s) is/are described with reference to the flowchartsillustrated in FIGS. 4-6, many other methods of implementing the exampleMDE 302 may alternatively be used. For example, the order of executionof the blocks may be changed, and/or some of the blocks described may bechanged, eliminated, or combined.

As mentioned above, the example processes of FIGS. 4-6 may beimplemented using coded instructions (e.g., computer and/or machinereadable instructions) stored on a tangible computer readable storagemedium such as a hard disk drive, a flash memory, a read-only memory(ROM), a compact disk (CD), a digital versatile disk (DVD), a cache, arandom-access memory (RAM) and/or any other storage device or storagedisk in which information is stored for any duration (e.g., for extendedtime periods, permanently, for brief instances, for temporarilybuffering, and/or for caching of the information). As used herein, theterm tangible computer readable storage medium is expressly defined toinclude any type of computer readable storage device and/or storage diskand to exclude propagating signals and to exclude transmission media. Asused herein, “tangible computer readable storage medium” and “tangiblemachine readable storage medium” are used interchangeably. Additionallyor alternatively, the example processes of FIGS. 4-6 may be implementedusing coded instructions (e.g., computer and/or machine readableinstructions) stored on a non-transitory computer and/or machinereadable medium such as a hard disk drive, a flash memory, a read-onlymemory, a compact disk, a digital versatile disk, a cache, arandom-access memory and/or any other storage device or storage disk inwhich information is stored for any duration (e.g., for extended timeperiods, permanently, for brief instances, for temporarily buffering,and/or for caching of the information). As used herein, the termnon-transitory computer readable medium is expressly defined to includeany type of computer readable storage device and/or storage disk and toexclude propagating signals and to exclude transmission media. As usedherein, when the phrase “at least” is used as the transition term in apreamble of a claim, it is open-ended in the same manner as the term“comprising” is open ended.

The program 400 of FIG. 4 begins at block 402 where the examplemarketing data interface 308 retrieves and/or otherwise receives a dataset having an unknown distribution of marketing data. Such data set(s)may be retrieved from one or more campaign data sources 304 via theexample network 306, in which the data set(s) include campaign resultsassociated with one or more promotions. In some examples, the dataset(s) include information related to a campaign population size andcorresponding information to indicate which ones of the campaignpopulation were exposed to promotional media (e.g., billboardadvertisements, television advertisements, Internet advertisements,etc.). The example GRP engine 310 calculates a GRP value associated withthe empirically derived data set, and the example reach engine 312calculates a corresponding reach value (block 404).

While the empirically determined GRP and reach values provide the marketanalyst with a measurement to indicate a number of impressions thepopulation experienced, and an indication of how many of thoseimpressions were unique, the market analyst also seeks to know how thereach will be affected in the event a new GRP value (published GRP) isachieved for that defined population. As described above, the marketanalyst may identify and/or otherwise select a new/published GRP valuethat can be targeted to the defined population. In some examples, themarket analyst can apply marketing resources to the defined population(e.g., increased advertising budget, additional media advertisements,etc.) with an expectation that a resulting published GRP value isachieved. However, because the data set associated with the definedpopulation reflects empirical data associated with the originalcampaign, any determination of a new/published reach value in view ofthe published GRP value must be predicted and/or otherwise estimated. Asdescribed above, conventional techniques to predict the published reachutilize computationally intensive nonlinear techniques to achieveconvergence when applying the NBD. Further, particular circumstances ofthe NBD cause computational failures, despite real-world inputs that arepractically observed in market behavior.

The example simulation engine 314 selects and/or otherwise retrieves apublished GRP of interest (block 406), and the example maximum entropyengine 318 estimates an unknown distribution (e.g., the empirical marketdata set) to determine a maximum entropy distribution (block 408).

FIG. 5 includes additional detail associated with determining themaximum entropy distribution of block 408. In the illustrated example ofFIG. 5, the example maximum entropy constraint manager 322 establishesconstraints for the unknown distribution to prevent computational waste(block 502), which may be achieved in a manner consistent with exampleEquations 13 and 14 described above. In particular, the maximum entropyconstraint manager 322 prevents computational waste by ensuring thatonly positive values are produced for a probability set, therebyavoiding potential impossibility region(s), such as those that produceerrors in an NBD model (see region 208 of FIG. 2). The example maximumentropy distribution evaluator 324 calculates a maximum entropy to theunknown distribution (block 504) in a manner consistent with exampleEquation 15, and the example maximum entropy constraint manager 322calculates zero-modified geometric distribution constraints tofacilitate cross entropy calculations (block 506), as described infurther detail below.

Returning to FIG. 4, the example minimum cross entropy engine 320modifies the calculated zero-modified geometric distribution nowrepresenting a maximum entropy condition with minimum cross entropy tocreate a new distribution that is constrained by the published GRP ofinterest (block 410). FIG. 6 includes additional detail associated withapplying the minimum cross entropy of block 410. In the illustratedexample of FIG. 6, the example minimum cross entropy constraint manager326 establishes constraints for calculating the minimum cross entropy(block 602) in a manner consistent with example Equations 19 and 20.Unlike conventional techniques, the example minimum cross entropy engine320 applies the minimum cross entropy based on available data underconsideration rather than forcing such data to fit within apredetermined distribution. In particular, the example minimum crossentropy engine 320 applies the minimum cross entropy based on (a) thepreviously calculated maximum entropy distribution (Q) and (b) theunknown distribution to be solved (P) (block 604) in a manner consistentwith example Equation 21. Additionally, the example minimum crossentropy engine 320 integrates assistance values with zero-modifiedgeometric distribution constraints (block 606) in a manner consistentwith example Equations 22 and 23. The example published GRP manager 328solves for the assistance values (e.g., s1) to facilitate theintegration of the published GRP constraint of interest (block 608) in amanner consistent with example Equation 24. Additionally, the examplepublished GRP manager 328 integrates the published GRP constraint (block610) in a manner consistent with example Equation 25.

Returning to FIG. 4, the example conserved quantity engine 316 solvesfor the published reach (block 412) in a manner consistent with exampleEquations 26-28. In particular, because the conserved quantities aresimilar on both sides of example Equation 28, desired quantities ofinterest can be solved for via closed-form conserved quantity equivalentexpressions (block 414), which are less computationally rigorous thanconventional methods that employ the NBD.

FIG. 7 is a chart 700 to illustrate example solutions to determine apublished reach as a function of raw reach. In the illustrated exampleof FIG. 7, a given raw GRP value is 50 having a corresponding publishedGRP of 200. The example chart 700 includes an x-axis of raw reach 702and a y-axis of published reach 704. The example chart 700 illustrates(a) an NBD solution 706, (b) a maximum entropy solution 708, (c) an NBDsolution using finite correction 710, and (d) a maximum entropy solutionusing a finite domain 712. As described above, the NBD solution 706 andthe NBD solution using finite correction 710 break down at particularvalues of raw reach, as shown in the chart 700 as a breakdown region714. Such breakdowns, when encountered by traditional industry standardapproaches sometimes revert to a Poisson analysis, which are completelyindependent of any actual observations available. In fact, suchsolutions remain constant and independent of any changes to the rawreach value(s). Stated differently, the Poisson analysis discards anypanel data that may have been available during the analysis. On theother hand, examples disclosed herein that employ the maximum entropysolution 708 and the maximum entropy finite domain solution 712 do notbreak down and, instead, converge. Additionally, the solution for theNBD using finite correction (710) breaks down at a low raw reach region716. However, example solutions disclosed herein (e.g., the maximumentropy solution 708 and the maximum entropy solution using a finitedomain 712) corrects for break downs at both extremes of raw reach. Asthe raw reach approaches a lower (toward zero) boundary, examplesdisclosed herein converge toward the origin, while traditionaltechniques converge to erroneous non-zero values (e.g., predicting thatreach is approximately 40% when an accurate value is closer to 1%). Inother words, if zero people were watching television (raw value), then acorresponding reach value is also expected to be at or near zero, ratherthan the erroneous traditional technique that illustrates approximately40%.

FIG. 8 is a block diagram of an example processor platform 800 capableof executing the instructions of FIGS. 4-6 and/or the example pseudocode of Table 2 to implement the market data evaluator system 300 and,in particular, the MDE 302 of FIG. 3. The processor platform 800 can be,for example, a server, a personal computer, a mobile device (e.g., acell phone, a smart phone, a tablet such as an iPad™), a personaldigital assistant (PDA), an Internet appliance, a set top box, or anyother type of computing device.

The processor platform 800 of the illustrated example includes aprocessor 812. The processor 812 of the illustrated example is hardware.For example, the processor 812 can be implemented by one or moreintegrated circuits, logic circuits, microprocessors or controllers fromany desired family or manufacturer. In the illustrated example of FIG.8, the processor 812 includes one or more example processing cores 815configured via example instructions 832, which include the exampleinstructions of FIGS. 4-6 and pseudo code of Table 2 to implement theexample MDE 302 of FIG. 3.

The processor 812 of the illustrated example includes a local memory 813(e.g., a cache). The processor 812 of the illustrated example is incommunication with a main memory including a volatile memory 814 and anon-volatile memory 816 via a bus 818. The volatile memory 814 may beimplemented by Synchronous Dynamic Random Access Memory (SDRAM), DynamicRandom Access Memory (DRAM), RAMBUS Dynamic Random Access Memory (RDRAM)and/or any other type of random access memory device. The non-volatilememory 816 may be implemented by flash memory and/or any other desiredtype of memory device. Access to the main memory 814, 816 is controlledby a memory controller.

The processor platform 800 of the illustrated example also includes aninterface circuit 820. The interface circuit 820 may be implemented byany type of interface standard, such as an Ethernet interface, auniversal serial bus (USB), and/or a PCI express interface.

In the illustrated example, one or more input devices 822 are connectedto the interface circuit 820. The input device(s) 822 permit(s) a userto enter data and commands into the processor 812. The input device(s)can be implemented by, for example, an audio sensor, a microphone, acamera (still or video), a keyboard, a button, a mouse, a touchscreen, atrack-pad, a trackball, isopoint, a voice recognition system and/or anyother human-machine interface.

One or more output devices 824 are also connected to the interfacecircuit 820 of the illustrated example. The output devices 824 can beimplemented, for example, by display devices (e.g., a light emittingdiode (LED), an organic light emitting diode (OLED), a liquid crystaldisplay, a cathode ray tube display (CRT), a touchscreen, a tactileoutput device, a printer and/or speakers). The interface circuit 820 ofthe illustrated example, thus, typically includes a graphics drivercard, a graphics driver chip or a graphics driver processor.

The interface circuit 820 of the illustrated example also includes acommunication device such as a transmitter, a receiver, a transceiver, amodem and/or network interface card to facilitate exchange of data withexternal machines (e.g., computing devices of any kind) via a network826 (e.g., an Ethernet connection, a digital subscriber line (DSL), atelephone line, coaxial cable, a cellular telephone system, etc.).

The processor platform 800 of the illustrated example also includes oneor more mass storage devices 828 for storing software and/or data.Examples of such mass storage devices 828 include floppy disk drives,hard drive disks, compact disk drives, Blu-ray disk drives, RAIDsystems, and digital versatile disk (DVD) drives.

The coded instructions 832 of FIGS. 4-6 and/or pseudo code of Table 2may be stored in the mass storage device 828, in the volatile memory814, in the non-volatile memory 816, and/or on a removable tangiblecomputer readable storage medium such as a CD or DVD.

From the foregoing, it will be appreciated that the above disclosedmethods, apparatus and articles of manufacture overcome computationallyintensive processing of systems that calculate reach values for marketdata. Additionally, examples disclosed herein eliminate inherentlimitations of conventional industry standard techniques whencalculating published reach values in connection with published GRPvalues, particularly in regard to applications of the NBD. Examplesdisclosed herein eliminate a need to perform iterative nonlineartechniques to produce convergence in market data distributions and,instead, facilitate closed-form expressions to calculate published GRP,published reach and/or published frequency estimates.

Although certain example methods, apparatus and articles of manufacturehave been disclosed herein, the scope of coverage of this patent is notlimited thereto. On the contrary, this patent covers all methods,apparatus and articles of manufacture fairly falling within the scope ofthe claims of this patent.

What is claimed is:
 1. A computer-implemented method to improve anefficiency of determining a published reach, comprising: identifying, byexecuting an instruction with at least one processor, a negativebinomial distribution feasibility region corresponding to a candidategross rating point (GRP) value; in response to identifying the negativebinomial distribution feasibility region is associated with a number ofsamples below a threshold, estimating, by executing an instruction withthe at least one processor, a sample distribution of marketing data togenerate a maximum entropy distribution, the maximum entropydistribution being constrained by a first GRP value and a first reachvalue, the first GRP value empirically measured, the first GRP valuecorresponding to the sample distribution of marketing data, the firstreach value based on the first GRP value; generating, by executing aninstruction with the at least one processor, a geometric distributionbased on estimating a minimum cross entropy of (a) the maximum entropydistribution and (b) the sample distribution of marketing data, theminimum cross entropy being constrained by the candidate GRP value ofthe sample distribution of marketing data, the candidate GRP value basedon an advertising campaign increase quantity; determining, by executingan instruction with the at least one processor, a model of the candidateGRP value, the model based on an assistance value corresponding to thecandidate GRP value; and reducing a computational burden associated withdetermining the published reach of the sample distribution of marketingdata by generating, by executing an instruction with the at least oneprocessor, closed-loop conserved quantity expressions of the geometricdistribution based on the model of the candidate GRP value.
 2. Thecomputer-implemented method as defined in claim 1, wherein the first GRPvalue and the first reach value are constrained for a probability ofzero advertising impressions associated with the sample distribution. 3.The computer-implemented method as defined in claim 1, whereinestimating the minimum cross entropy includes applying aKullback-Leibler divergence probability.
 4. The computer-implementedmethod as defined in claim 1, wherein the conserved quantity expressionsassociate at least one of (a) GRP and reach, (b) GRP and frequency, or(c) reach and frequency.
 5. An apparatus to improve an efficiency ofdetermining a published reach, comprising: a market data evaluator toidentify a negative binomial distribution feasibility regioncorresponding to a candidate gross rating point (GRP) value; a maximumentropy engine to: in response to the market data evaluator identifyingthe negative binomial distribution feasibility region is associated witha number of samples below a threshold: generate a maximum entropydistribution, based on estimating a sample distribution of marketingdata, the maximum entropy distribution constrained by a first GRP valueand a first reach value, the first GRP value corresponding to the sampledistribution of marketing data, the first reach value based on the firstGRP value; a maximum entropy constraint manager to generate a geometricdistribution based on estimating a minimum cross entropy of (a) themaximum entropy distribution and (b) the sample distribution ofmarketing data; a minimum cross entropy constraint manager to constrainthe minimum cross entropy with the candidate GRP value of the sampledistribution of marketing data, the candidate GRP value based on anadvertising campaign increase quantity; and a conserved quantity engineto: determine a model of the candidate GRP value, the model based on anassistance value corresponding to the candidate GRP value; and reduce acomputational burden associated with determining the published reach ofthe sample distribution of marketing data by generating closed-loopconserved quantity expressions of the geometric distribution based onthe model of the candidate GRP value.
 6. The apparatus as defined inclaim 5, wherein the maximum entropy engine is to constrain the firstGRP value and the first reach value for a probability of zeroadvertising impressions associated with the sample distribution.
 7. Theapparatus as defined in claim 5, further including a minimumcross-entropy engine to estimate the minimum cross entropy with aKullback-Leibler divergence probability.
 8. The apparatus as defined inclaim 5, wherein the conserved quantity expressions associate at leastone of (a) GRP and reach, (b) GRP and frequency, or (c) reach andfrequency.
 9. A tangible computer readable storage medium comprisinginstructions to improve an efficiency of determining a published reachthat, when executed, causes a processor to, at least: identify anegative binomial distribution feasibility region corresponding to acandidate gross rating point (GRP) value; in response to identifying thenegative binomial distribution feasibility region is associated with anumber of samples below a threshold, estimate a sample distribution ofmarketing data to generate a maximum entropy distribution, the maximumentropy distribution being constrained by a first GRP value and a firstreach value, the first GRP value empirically measured, the first GRPvalue corresponding to the sample distribution of marketing data, thefirst reach value based on the first GRP value; generate a geometricdistribution based on estimating a minimum cross entropy of (a) themaximum entropy distribution and (b) the sample distribution ofmarketing data, the minimum cross entropy being constrained by thecandidate GRP value of the sample distribution of marketing data, thecandidate GRP value based on an advertising campaign increase quantity;determine a model of the candidate GRP value, the model based on anassistance value corresponding to the candidate GRP value; and reduce acomputational burden associated with determining the published reach ofthe sample distribution of marketing data by generating closed-loopconserved quantity expressions of the geometric distribution based onthe model of the candidate GRP value.
 10. The tangible computer readablestorage medium of claim 9, wherein the instructions, when executed,further cause the processor to constrain the first GRP value and thefirst reach value for a probability of zero advertising impressionsassociated with the sample distribution.